Sarine draws a triangle. She measures the length of the sides and records her measurements as follows. What is the measure of angle C of the triangle?

### Law of Cosines with SSS

The Law of Cosines,

which can be further manipulated to

Let's find the measure of the largest angle in the triangle with side lengths 12, 18 and 21.

First, we must determine which angle will be the largest. Recall from Geometry that the longest side is opposite the largest angle. The longest side is 21 so we will let

**Note:** Be careful to put parenthesis around the entire numerator and entire denominator on the calculator to ensure the proper order of operations. Your calculator screen should look like this:

Now let's find the value of

The angle with measure

Finally, let's find the

First, let’s rearrange the formula to reflect the sides given and requested angle:

**Examples**

**Example 1**

Earlier, you were asked to find the measure of angle C of the triangle that has sides a = 3, b = 4, and c = 5.

We can use the manipulated Law of Cosines to solve for C.

Therefore, the triangle is a right triangle.

#### Example 2

Find the measure of

#### Example 3

Find the measure of the smallest angle in the triangle with side lengths 47, 54 and 72.

The smallest angle will be opposite the side with length 47, so this will be our

#### Example 4

Find

Rearrange the formula to solve for

### Review

Use the Law of Cosines to find the value of

- Find the measure of the smallest angle in the triangle with side lengths 150, 165 and 200 meters.
- Find the measure of the largest angle in the triangle with side length 59, 83 and 100 yards.
- Find the
m∠C ifa=6,b=9 andc=13 . - Find the
m∠B ifa=15,b=8 andc=9 . - Find the
m∠A ifa=24,b=20 andc=14 . - A triangular plot of land is bordered by a road, a fence and a creek. If the stretch along the road is 100 meters, the length of the fence is 115 meters and the side along the creek is 90 meters, at what angle do the fence and road meet?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 13.16.